Continuous Probability Distributions

3.3 Continuous Probability Distributions

A continuous random variable has a probability of 0 of assuming exactly any of its values. Consequently, its probability distribution cannot be given in tabular form.

88 Chapter 3 Random Variables and Probability Distributions At first this may seem startling, but it becomes more plausible when we consider a

particular example. Let us discuss a random variable whose values are the heights of all people over 21 years of age. Between any two values, say 163.5 and 164.5 centimeters, or even 163.99 and 164.01 centimeters, there are an infinite number of heights, one of which is 164 centimeters. The probability of selecting a person at random who is exactly 164 centimeters tall and not one of the infinitely large set of heights so close to 164 centimeters that you cannot humanly measure the difference is remote, and thus we assign a probability of 0 to the event. This is not the case, however, if we talk about the probability of selecting a person who is at least 163 centimeters but not more than 165 centimeters tall. Now we are dealing with an interval rather than a point value of our random variable.

We shall concern ourselves with computing probabilities for various intervals of continuous random variables such as P (a < X < b), P (W ≥ c), and so forth. Note that when X is continuous,

P (a < X ≤ b) = P (a < X < b) + P (X = b) = P (a < X < b). That is, it does not matter whether we include an endpoint of the interval or not. This is not true, though, when X is discrete.

Although the probability distribution of a continuous random variable cannot

be presented in tabular form, it can be stated as a formula. Such a formula would necessarily be a function of the numerical values of the continuous random variable

X and as such will be represented by the functional notation f (x). In dealing with continuous variables, f (x) is usually called the probability density function, or simply the density function, of X. Since X is defined over a continuous sample space, it is possible for f (x) to have a finite number of discontinuities. However, most density functions that have practical applications in the analysis of statistical data are continuous and their graphs may take any of several forms, some of which are shown in Figure 3.4. Because areas will be used to represent probabilities and probabilities are positive numerical values, the density function must lie entirely above the x axis.

(a)

(d) Figure 3.4: Typical density functions.

(b)

(c)

A probability density function is constructed so that the area under its curve

3.3 Continuous Probability Distributions

89 bounded by the x axis is equal to 1 when computed over the range of X for which

f (x) is defined. Should this range of X be a finite interval, it is always possible to extend the interval to include the entire set of real numbers by defining f (x) to

be zero at all points in the extended portions of the interval. In Figure 3.5, the probability that X assumes a value between a and b is equal to the shaded area under the density function between the ordinates at x = a and x = b, and from integral calculus is given by

P (a < X < b) =

f (x) dx.

f(x) a

Figure 3.5: P (a < X < b).

Definition 3.6: The function f (x) is a probability density function (pdf) for the continuous random variable X, defined over the set of real numbers, if

Example 3.11: Suppose that the error in the reaction temperature, in ◦

C, for a controlled labora- tory experiment is a continuous random variable X having the probability density function

. (a) Verify that f (x) is a density function. (b) Find P (0 < X ≤ 1).

Solution : We use Definition 3.6. (a) Obviously, f (x) ≥ 0. To verify condition 2 in Definition 3.6, we have

2 8 f (x) dx = 1 dx = | −1 = + = 1.

90 Chapter 3 Random Variables and Probability Distributions (b) Using formula 3 in Definition 3.6, we obtain

P (0 < X ≤ 1) =

dx =

0 3 9 0 9 Definition 3.7: The cumulative distribution function F (x) of a continuous random variable

X with density function f (x) is

As an immediate consequence of Definition 3.7, one can write the two results dF (x)

P (a < X < b) = F (b) − F (a) and f(x) =

, dx

if the derivative exists. Example 3.12: For the density function of Example 3.11, find F (x), and use it to evaluate

P (0 < X ≤ 1). Solution : For −1 < x < 2,

The cumulative distribution function F (x) is expressed in Figure 3.6. Now

2 1 1 P (0 < X ≤ 1) = F (1) − F (0) = − = ,

9 9 9 which agrees with the result obtained by using the density function in Example

3.11. Example 3.13: The Department of Energy (DOE) puts projects out on bid and generally estimates

what a reasonable bid should be. Call the estimate b. The DOE has determined that the density function of the winning (low) bid is

Find F (y) and use it to determine the probability that the winning bid is less than the DOE’s preliminary estimate b.

Solution : For 2b/5 ≤ y ≤ 2b,

2b/5 8b 8b 2b/5

8b 4

Figure 3.6: Continuous cumulative distribution function.

To determine the probability that the winning bid is less than the preliminary bid estimate b, we have

5 1 3 P (Y ≤ b) = F (b) = − = .

Exercises

3.1 Classify the following random variables as dis- then to each sample point assign a value x of the ran- crete or continuous:

dom variable X representing the number of automo- X: the number of automobile accidents per year biles with paint blemishes purchased by the agency. in Virginia.

3.3 Let W be a random variable giving the number Y : the length of time to play 18 holes of golf.

of heads minus the number of tails in three tosses of a M : the amount of milk produced yearly by a par- coin. List the elements of the sample space S for the ticular cow.

three tosses of the coin and to each sample point assign

a value w of W .

N : the number of eggs laid each month by a hen. 3.4 A coin is flipped until 3 heads in succession oc- P : the number of building permits issued each cur. List only those elements of the sample space that

month in a certain city. require 6 or less tosses. Is this a discrete sample space? Q: the weight of grain produced per acre.

Explain.

3.5 Determine the value c so that each of the follow- 3.2 An overseas shipment of 5 foreign automobiles ing functions can serve as a probability distribution of

contains 2 that have slight paint blemishes. If an the discrete random variable X: agency receives 3 of these automobiles at random, list

the elements of the sample space S, using the letters B (a) f (x) = c(x + 4), for x = 0, 1, 2, 3; and N for blemished and nonblemished, respectively; (b) f (x) = c 2 x 3 3−x , for x = 0, 1, 2.

92 Chapter 3 Random Variables and Probability Distributions 3.6 The shelf life, in days, for bottles of a certain

3.12 An investment firm offers its customers munici- prescribed medicine is a random variable having the pal bonds that mature after varying numbers of years. density function

Given that the cumulative distribution function of T , the number of years to maturity for a randomly se-

(x+100) 3 , x > 0,

lected bond, is

Find the probability that a bottle of this medicine will

1 ≤ t < 3, have a shell life of

3 ≤ t < 5, (a) at least 200 days;

F (t) =

5 ≤ t < 7, (b) anywhere from 80 to 120 days.

⎩ 1, t ≥ 7, 3.7 The total number of hours, measured in units of find

100 hours, that a family runs a vacuum cleaner over a (a) P (T = 5); period of one year is a continuous random variable X (b) P (T > 3); that has the density function

(c) P (1.4 < T < 6);

3.13 The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in Find the probability that over a period of one year, a continuous rolls of uniform width, is given by family runs their vacuum cleaner

elsewhere.

0 1 2 3 4 (a) less than 120 hours;

0.41 0.37 0.16 0.05 0.01 (b) between 50 and 100 hours.

f (x)

Construct the cumulative distribution function of X. 3.14 The waiting time, in hours, between successive

3.8 Find the probability distribution of the random speeders spotted by a radar unit is a continuous ran- variable W in Exercise 3.3, assuming that the coin is dom variable with cumulative distribution function biased so that a head is twice as likely to occur as a tail.

x < 0,

3.9 The proportion of people who respond to a certain 1−e −8x , x ≥ 0. mail-order solicitation is a continuous random variable

F (x) =

X that has the density function Find the probability of waiting less than 12 minutes between successive speeders

(a) using the cumulative distribution function of X; f (x) =

2(x+2)

0 < x < 1,

(b) using the probability density function of X. (a) Show that P (0 < X < 1) = 1.

elsewhere.

3.15 Find the cumulative distribution function of the (b) Find the probability that more than 1/4 but fewer random variable X representing the number of defec-

than 1/2 of the people contacted will respond to tives in Exercise 3.11. Then using F (x), find this type of solicitation.

(a) P (X = 1); (b) P (0 < X ≤ 2).

3.10 Find a formula for the probability distribution of the random variable X representing the outcome when

3.16 Construct a graph of the cumulative distribution a single die is rolled once.

function of Exercise 3.15.

3.11 A shipment of 7 television sets contains 2 de- 3.17 A continuous random variable X that can as- fective sets. A hotel makes a random purchase of 3 sume values between x = 1 and x = 3 has a density of the sets. If x is the number of defective sets pur- function given by f (x) = 1/2. chased by the hotel, find the probability distribution (a) Show that the area under the curve is equal to 1. of X. Express the results graphically as a probability histogram.

(b) Find P (2 < X < 2.5).

(c) Find P (X ≤ 1.6).

Exercises

93 3.18 A continuous random variable X that can as- (a) Find F (x).

sume values between x = 2 and x = 5 has a density (b) Determine the probability that the component (and function given by f (x) = 2(1 + x)/27. Find

thus the DVD player) lasts more than 1000 hours (a) P (X < 4);

before the component needs to be replaced. (b) P (3 ≤ X < 4).

(c) Determine the probability that the component fails

before 2000 hours.

3.19 For the density function of Exercise 3.17, find F (x). Use it to evaluate P (2 < X < 2.5).

3.28 A cereal manufacturer is aware that the weight of the product in the box varies slightly from box 3.20 For the density function of Exercise 3.18, find to box. In fact, considerable historical data have al- F (x), and use it to evaluate P (3 ≤ X < 4).

lowed the determination of the density function that describes the probability structure for the weight (in

3.21 Consider the density function ounces). Letting X be the random variable weight, in ounces, the density function can be described as

0, elsewhere. (a) Evaluate k.

(a) Verify that this is a valid density function. (b) Find F (x) and use it to evaluate

(b) Determine the probability that the weight is

smaller than 24 ounces.

P (0.3 < X < 0.6). (c) The company desires that the weight exceeding 26

ounces be an extremely rare occurrence. What is 3.22 Three cards are drawn in succession from a deck

the probability that this rare occurrence does ac- without replacement. Find the probability distribution

tually occur?

for the number of spades. 3.29 An important factor in solid missile fuel is the 3.23 Find the cumulative distribution function of the particle size distribution. Significant problems occur if random variable W in Exercise 3.8. Using F (w), find

the particle sizes are too large. From production data (a) P (W > 0);

in the past, it has been determined that the particle size (in micrometers) distribution is characterized by

(b) P (−1 ≤ W < 3). f (x) = 3x −4 , x > 1, 3.24 Find the probability distribution for the number

elsewhere. of jazz CDs when 4 CDs are selected at random from a collection consisting of 5 jazz CDs, 2 classical CDs, (a) Verify that this is a valid density function. and 3 rock CDs. Express your results by means of a (b) Evaluate F (x). formula.

(c) What is the probability that a random particle from the manufactured fuel exceeds 4 micrometers?

3.25 From a box containing 4 dimes and 2 nickels, 3 coins are selected at random without replacement.

3.30 Measurements of scientific systems are always Find the probability distribution for the total T of the subject to variation, some more than others. There

3 coins. Express the probability distribution graphi- are many structures for measurement error, and statis- cally as a probability histogram.

ticians spend a great deal of time modeling these errors. Suppose the measurement error X of a certain physical

3.26 From a box containing 4 black balls and 2 green quantity is decided by the density function balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find

k(3 − x 2 ), −1 ≤ x ≤ 1, the probability distribution for the number of green

f (x) =

elsewhere. balls. (a) Determine k that renders f (x) a valid density func-

3.27 The time to failure in hours of an important

tion.

piece of electronic equipment used in a manufactured (b) Find the probability that a random error in mea- DVD player has the density function

surement is less than 1/2. 1 (c) For this particular measurement, it is undesirable

f (x) = 2000 exp(−x/2000), x ≥ 0, 0,

if the magnitude of the error (i.e., |x|) exceeds 0.8. What is the probability that this occurs?

x < 0.

94 Chapter 3 Random Variables and Probability Distributions 3.31 Based on extensive testing, it is determined by

3.34 Magnetron tubes are produced on an automated the manufacturer of a washing machine that the time assembly line. A sampling plan is used periodically to Y (in years) before a major repair is required is char- assess quality of the lengths of the tubes. This mea- acterized by the probability density function

surement is subject to uncertainty. It is thought that 1 the probability that a random tube meets length spec-

f (y) = 4 e −y/4 , y ≥ 0, ification is 0.99. A sampling plan is used in which the 0,

lengths of 5 random tubes are measured. (a) Critics would certainly consider the product a bar- (a) Show that the probability function of Y , the num-

elsewhere.

gain if it is unlikely to require a major repair before ber out of 5 that meet length specification, is given the sixth year. Comment on this by determining

by the following discrete probability function: P (Y > 6).

(0.99) y (0.01) 5−y , in the first year?

(b) What is the probability that a major repair occurs

f (y) =

y!(5 − y)!

for y = 0, 1, 2, 3, 4, 5.

3.32 The proportion of the budget for a certain type (b) Suppose random selections are made off the line of industrial company that is allotted to environmental

and 3 are outside specifications. Use f (y) above ei- and pollution control is coming under scrutiny. A data

ther to support or to refute the conjecture that the collection project determines that the distribution of

probability is 0.99 that a single tube meets specifi- these proportions is given by

3.35 Suppose it is known from large amounts of his- torical data that X, the number of cars that arrive at

(a) Verify that the above is a valid density function. a specific intersection during a 20-second time period, is characterized by the following discrete probability

(b) What is the probability that a company chosen at function: random expends less than 10% of its budget on en- vironmental and pollution controls?

f (x) = e −6 6 (c) What is the probability that a company selected x , for x = 0, 1, 2, . . . .

x!

at random spends more than 50% of its budget on environmental and pollution controls?

(a) Find the probability that in a specific 20-second time period, more than 8 cars arrive at the

3.33 Suppose a certain type of small data processing

intersection.

firm is so specialized that some have difficulty making (b) Find the probability that only 2 cars arrive. a profit in their first year of operation. The probabil- ity density function that characterizes the proportion

3.36 On a laboratory assignment, if the equipment is Y that make a profit is given by

working, the density function of the observed outcome,

f (x) = 2(1 − x), 0 < x < 1, (a) What is the value of k that renders the above a

elsewhere.

otherwise. valid density function? (b) Find the probability that at most 50% of the firms (a) Calculate P (X ≤ 1/3). make a profit in the first year.

(b) What is the probability that X will exceed 0.5? (c) Find the probability that at least 80% of the firms (c) Given that X ≥ 0.5, what is the probability that

make a profit in the first year.

X will be less than 0.75?